I'm dubious on the issue of "hypermobility". That's not a diagnosis and doesn't help you. Actual instability is something else entirely, but requires x-ray to really see and you would likely have neurological symptoms.
As usual, internet doctoring is difficult because we only get a piece of the picture but it sounds to be a facet or disc generated pain in your lower back. I don't have any specific advice on that issue because I haven't examined you, but I doubt that the yoga is necessarily bad for you. If it causes pain it should be avoided for the time being, to be re-visited later on when you are feeling better. My best advice is to leave the system and seek out a chiropractor and and acupuncturist to help with your lower back issues if you are not getting the help you need.
For the shin issues, be diligent about working out the sore muscles daily. Start at the upper part of just inside your shin and press. If there is pain, hold it with moderate pressure until the pain subsides, then move down a thumb width and repeat. Use a rolling pin on your calves both sides and the bottom of your feet, again both sides. I usually suggest three minutes, twice a day for the calves and feet. Do the shin treatment twice a day too.
To answer your question on BF running. You don't need to stop just because. Listen to your body. If you hurt, slow down. Go for a walk if running hurts. If I go out for a run and it doesn't feel right, I slow down. If it still doesn't feel right, I go home and try again tomorrow. Remember Caballo's mantra: easy, light, smooth, then fast. Read a lot on this forum, and there are plenty of books out there that will help. Arches too high? Nonsense. They are arches.
I think the best thing you can do is find someone who will work with you, instead of pass you off.